Quick answer

Find a = -1/m, build y = ax + b through (x0, y0), solve mx + r = ax + b.

Formula

  • a = -1/m
  • b = y0 - a·x0
  • x = (b - r)/(m - a)

Introduction

Manual calculation builds confidence before you rely on tools. Use the calculator on the home page to verify each step when fractions get long.

This walkthrough assumes you already understand why slopes multiply to -1. If not, open the perpendicular line formula page first for the symbolic rules.

We use the common classroom setup: a base line y = mx + r and a point (x0, y0) that the perpendicular must pass through.

After you master the sequence, work through numbered cases in the perpendicular line examples article to build speed before a quiz.

What you need before starting

Know the base line in slope-intercept form and one point the perpendicular must pass through.

Keep fractions exact until the final answer unless the problem requests decimals.

Label every value on your paper before substituting. Mixing up base slope and perpendicular slope is the most common error.

Have graph paper or graphing software nearby. A quick plot catches mistakes that algebra alone might hide.

Formula and relationships

  • a = -1/m
  • b = y0 - a·x0
  • x = (b - r)/(m - a)

Label m, r, x0, and y0 before substituting to avoid mix-ups.

The intersection formula x = (b - r)/(m - a) comes directly from equating the two expressions for y.

When m = a, the lines are parallel and the perpendicular construction failed a slope check.

Step-by-step guide

  1. Read m and r. From y = mx + r on the diagram or in the prompt.
  2. Find a. Use a = -1/m, or x = x0 when the base is horizontal.
  3. Find b. Substitute the point into y = ax + b.
  4. Write the perpendicular equation. Box the final line so graders can find it quickly.
  5. Solve for intersection. Set mx + r equal to ax + b and solve for x.
  6. Back-substitute for y. Use either line; both must give the same coordinate.
  7. Check. Multiply slopes to see -1 when both are defined.

Worked examples

Base line y = x + 2 and perpendicular through (0, 3) give a = -1 and y = -x + 3.

Intersection: x + 2 = -x + 3 leads to x = 0.5 and y = 2.5.

A second check: slopes 1 and -1 multiply to -1, which supports the right-angle claim.